There is an alcohol rehab center in Princess Anne, a town in Maryland, USA (founded 1733). Other than that, I've never heard the word "alcohol" in relation to the present Anne, Princess Royal, who does more working engagements than any of the other Royals, sometimes several per day. It would be hard to do that, or her renowned horsewoman pursuits, while drunk.
Do you have a link for your claim?
He may be confusing Princess Anne with Princess Margaret. Margaret was a heavy drinker.